Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of finite-dimensional representations of $G$ to the category $\mathrm{Rep}^f(\mathfrak g)$ of finite-dimensional representations of $\mathfrak g$ is an equivalence of categories. The inverse functor deserves to be called "integration".

There are actually two versions of this theorem, one for real (necessarily analytic) groups and real representations, and one for complex (holomorphic) groups and complex representations. In both cases the proof is essentially no worse than the existence of solutions to ODEs.

My question is about "Lie monoids", which are monoid objects in (real or complex) manifolds that are not necessarily groups. Any Lie monoid $M$ has a maximal sub-Lie group $M^\times$ (its group of invertible elements), and the infinitesimal neighborhood of $e\in M$ certainly cannot see outside $M^\times$. Thus one can define $\mathrm{Lie}(M) = \mathrm{Lie}(M^\times)$. Note that, unlike in the group case, this is no longer the same as the Lie algebra of left-translation-invariant vector fields. I do still have a "differentiation" functor $\mathrm{Rep}^f(M) \to \mathrm{Rep}^f(\mathrm{Lie}(M))$.

Are there conditions on $M$, short of demanding that it be a group, in which I might still have an equivalence of categories?

What if instead I used the Lie algebra of left-translation-invariant vector fields?

Addendum: The comments have brought up a lot of interesting mathematics, but also suggest that perhaps I didn't make my question clear. Here is a version of what I meant to ask:

Among the monoids $M$ with a given Lie algebra $\mathfrak g$, some of them have the property that $\mathrm{Rep}^f(M) \to \mathrm{Rep}^f(\mathfrak g)$ is an equivalence. (Provided $M$ is connected, I think it suffices for every $\mathfrak g$-module to extend to an $M$-module.) This class contains all the connected simply-connected groups with Lie algebra $\mathfrak g$ (of which there is, of course, only one up to unique isomorphism). It is not limited just to the connected simply-connected groups: if $\mathfrak g = \mathfrak{sl}(2,\mathbb R)$, then this class of monoids contains $\mathrm{SL}(2,\mathbb R)$, for example. Might it also contain non-group monoids?

finite-dimensional, apparently? $\endgroup$ – Allen Knutson May 5 '15 at 5:22